Related papers: A two-dimensional polynomial mapping with a wander…

We determine all couples of commuting polynomial endomorphisms of C^2 that extends to holomorphic endomorphisms of P^2 and that have disjoint sequences of iterates.

We study invariant Fatou components for holomorphic endomorphisms in $\mathbb{P}^2$. In the recurrent case these components were classified by Sibony and the second author in 1995. In 2008 Ueda completed this classification by proving that…

We study the iteration of transcendental self-maps of $\mathbb{C}^*:=\mathbb{C}\setminus \{0\}$, that is, holomorphic functions $f:\mathbb{C}^*\to\mathbb{C}^*$ for which both zero and infinity are essential singularities. We use…

In this paper, we show that there exist transcendental meromorphic functions with a cycle of 2-periodic Fatou components, where one is simply connected while the other is doubly connected. In particular, the doubly connected Fatou component…

This note initiates the study of the Fatou\,--\,Julia sets of a complex harmonic mapping. Along with some fundamental properties of the Fatou and the Julia sets, we observe some contrasting behaviour of these sets as those with in case of a…

The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can…

Let $f$ be a transcendental entire function and $U$ be a Fatou component of $f$. We show that if $U$ is an escaping wandering domain of $f$, then most boundary points of $U$ (in the sense of harmonic measure) are also escaping. In the other…

A {\sl parabolic cylinder} is an invariant, non-recurrent Fatou component $\Omega$ of an automorphism $F$ of $\mathbb C^2$ satisfying: (1) The closure of the $\omega$-limit set of $F$ on $\Omega$ contains an isolated fixed point, (2) there…

We give an estimate of the discontinuity of the large Julia set for a perturbation of a class of maps tangent to the identity, by means of a two-dimensional Lavaurs Theorem. We adapt to our situation a strategy due to Bedford, Smillie and…

We consider polynomial maps of the form f(z,w) = (p(z),q(z,w)) that extend as holomorphic maps of CP^2. Mattias Jonsson introduces in (Math. Ann., 1999) a notion of connectedness for such polynomial skew products that is analogous to…

We describe conditions under which a multiply connected wandering domain of a transcendental meromorphic function with a finite number of poles must be a Baker wandering domain, and we discuss the possible eventual connectivity of Fatou…

We prove several results concerning the relative position of points in the postsingular set $P(f)$ of a meromorphic map $f$ and the boundary of a Baker domain or the successive iterates of a wandering component. For Baker domains we answer…

Let $X$ be a closed, $1$-dimensional, complex subvariety of $\CC^2$ and let $\ol{\BB}$ be a closed ball in $\CC^2 - X$. Then there exists a Fatou-Bieberbach domain $\Omega$ with $X \subseteq \Omega \subseteq \CC^2 - \ol{\BB}$ and a…

This paper continues our investigation of the dynamics of polynomial diffeomorphisms of C^2. We introduce a dynamical property of polynomial diffeomorphisms that generalizes hyperbolicity in the way that semi-hyperbolicity generalizes…

Short $\mathbb{C}^2$'s were constructed in [F] as attracting basins of a sequence of holomorphic automorphisms whose rate of attraction increases superexponentially. The goal of this paper is to show that such domains also arise naturally…

We prove the existence of automorphisms of $\mathbb C^k$, $k\ge 2$, having an invariant, non-recurrent Fatou component biholomorphic to $\mathbb C \times (\mathbb C^\ast)^{k-1}$ which is attracting, in the sense that all the orbits converge…

We study the dynamics of polynomial mappings f:C^k to C^k of degree at least 2 that extend continuously to projective space P^k. Our approach is to study the dynamics near the hyperplane at infinity and then making a descent to K --- the…

We analyze the boundaries of multiply connected Fatou components of transcendental maps by means of universal covering maps and associated inner functions. A unified approach is presented, which includes invariant Fatou components (of any…

In \cite{Bedford}, the dynamics of a particular polynomial diffeomorphism of $\mathbb{C}^N$, called a polynomial shift-like map of type $\nu$, has been studied as a higher dimensional analog of H\'enon maps. In this note, we prove that the…

We extend a recent result on the existence of wandering domains of polynomial functions defined over the p-adic field C_p to any algebraically closed complete non-archimedean field C_K with residue characteristic p>0. We also prove that…